Transcript 10-8 Counting Principles

10-8 Counting Principles
Warm Up
Problem of the Day
Lesson Presentation
Course 3
10-8 Counting Principles
Warm Up
An experiment consists of rolling a fair
number cube with faces numbered 2, 4, 6, 8,
10, and 12. Find each probability.
1. P(rolling an even number) 1
2. P(rolling a prime number) 1
3. P(rolling a number > 7)
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1
2
10-8 Counting Principles
Problem of the Day
There are 10 players in a chess
tournament. How many games are
needed for each player to play every other
player one time?
45
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10-8 Counting Principles
Learn to find the number of possible
outcomes in an experiment.
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Principles
10-8 Counting
Insert Lesson
Title Here
Vocabulary
Fundamental Counting Principle
tree diagram
Addition Counting Principle
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10-8 Counting Principles
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10-8 Counting Principles
Additional Example 1A: Using the Fundamental
Counting Principle
License plates are being produced that have a
single letter followed by three digits. All
license plates are equally likely.
Find the number of possible license plates.
Use the Fundamental Counting Principal.
letter
first digit
second digit
third digit
26 choices 10 choices 10 choices 10 choices
26 • 10 • 10 • 10 = 26,000
The number of possible 1-letter, 3-digit license
plates is 26,000.
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10-8 Counting Principles
Additional Example 1B: Using the Fundamental
Counting Principal
Find the probability that a license plate
has the letter Q.
P(Q
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1
) = 1 • 10 • 10 • 10 =
 0.038
26,000
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10-8 Counting Principles
Additional Example 1C: Using the Fundamental
Counting Principle
Find the probability that a license plate does not
contain a 3.
First use the Fundamental Counting Principle to
find the number of license plates that do not
contain a 3.
26 • 9 • 9 • 9 = 18,954 possible license plates
without a 3
There are 9 choices for any
digit except 3.
P(no 3) = 18,954 = 0.729
26,000
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10-8 Counting Principles
Check It Out: Example 1A
Social Security numbers contain 9 digits. All
social security numbers are equally likely.
Find the number of possible Social Security
numbers.
Use the Fundamental Counting Principle.
Digit
1
Choices 10
2
10
3
4
10 10
5
10
6
7
10 10
8
9
10
10
10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 =
1,000,000,000
The number of Social Security numbers is
1,000,000,000.
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10-8 Counting Principles
Check It Out: Example 1B
Find the probability that the Social Security
number contains a 7.
P(7 _ _ _ _ _ _ _ _) = 1 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10
1,000,000,000
=
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1 = 0.1
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10-8 Counting Principles
Check It Out: Example 1C
Find the probability that a Social Security
number does not contain a 7.
First use the Fundamental Counting Principle to find
the number of Social Security numbers that do not
contain a 7.
P(no 7 _ _ _ _ _ _ _ _) = 9 • 9 • 9 • 9 • 9 • 9 • 9 • 9 • 9
1,000,000,000
P(no 7) =
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387,420,489
1,000,000,000
≈ 0.4
10-8 Counting Principles
The Fundamental Counting Principle tells you
only the number of outcomes in some
experiments, not what the outcomes are. A tree
diagram is a way to show all of the possible
outcomes.
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10-8 Counting Principles
Additional Example 2: Using a Tree Diagram
You have a photo that you want to mat and
frame. You can choose from a blue, purple, red,
or green mat and a metal or wood frame.
Describe all of the ways you could frame this
photo with one mat and one frame.
You can find all of the possible outcomes by
making a tree diagram.
There should be 4 • 2 = 8 different ways to
frame the photo.
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10-8 Counting Principles
Additional Example 2 Continued
Each “branch” of the tree
diagram represents a
different way to frame the
photo. The ways shown in
the branches could be
written as (blue, metal),
(blue, wood), (purple,
metal), (purple, wood),
(red, metal), (red, wood),
(green, metal), and (green,
wood).
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10-8 Counting Principles
Check It Out: Example 2
A baker can make yellow or white cakes
with a choice of chocolate, strawberry, or
vanilla icing. Describe all of the possible
combinations of cakes.
You can find all of the possible outcomes by
making a tree diagram.
There should be 2 • 3 = 6 different cakes
available.
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10-8 Counting Principles
Check It Out: Example 2 Continued
yellow cake
vanilla icing
chocolate
icing
strawberry
icing
white cake
vanilla icing
chocolate
icing
strawberry
icing
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The different cake
possibilities are
(yellow, chocolate),
(yellow, strawberry),
(yellow, vanilla),
(white, chocolate),
(white, strawberry),
and (white, vanilla).
10-8 Counting Principles
Additional Example 3: Using the Addition Counting
Principle
The table shows the items available at a farm
stand. How many items can you choose from
the farm stand?
Apples
Pears
Squash
Macintosh
Bosc
Acorn
Red Delicious
Yellow Bartlett
Hubbard
Gold Delicious
Red Bartlett
None of the lists contains identical items, so use
the Addition Counting Principle.
Total Choices
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= Apples +
Pears + Squash
10-8 Counting Principles
Additional Example 3 Continued
T
=
3
+
3
+
2
There are 8 items to choose from.
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=8
10-8 Counting Principles
Check It Out: Example 3
The table shows the items available at a
clothing store. How many items can you
choose from the clothing store?
T-Shirts
Sweaters
Pants
Long Sleeve
Wool
Denim
Shirt Sleeve
Cotton
Khaki
Pocket
Polyester
Cashmere
None of the lists contains identical items, so use
the Addition Counting Principle.
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10-8 Counting Principles
Additional Example 3 Continued
Total Choices
T
=
= T-shirts + Sweaters +
3
+
4
+
2
There are 9 items to choose from.
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Pants
=9
Principles
10-8 Counting
Insert Lesson
Title Here
Lesson Quiz: Part I
Personal identification numbers (PINs)
contain 2 letters followed by 4 digits. Assume
that all codes are equally likely.
1. Find the number of possible PINs. 6,760,000
2. Find the probability that a PIN does not contain
0.6561
a 6.
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Principles
10-8 Counting
Insert Lesson
Title Here
Lesson Quiz: Part II
A lunch menu consists of 3 types of
sandwiches, 2 types of soup, and 3 types of
fruit.
3. What is the total number of lunch items on the
t menu? 8
4. A student wants to order one sandwich, one
t bowl of soup, and one piece of fruit. How many
t different lunches are possible?
18
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